Spin Chronicles Part 26: Imagine Clifford Algebra

 The song is in C major. Even if I like C minor better, I like the song.

"Imagine all the people

Livin' life in peace

You"

Can you imagine it? I can, even though it is not in my memory banks. Imagination is a powerful mechanism. I am reading "The Story of the Tower of Babel: Human Aspirations and Dreams".

There:

"Human imagination is a unique and powerful ability that allows us to conceive of things that do not exist, create new concepts, explore possibilities, and even transcend the current physical limitations in spirit. This ability is indeed rare in the natural world, and, at least based on our current knowledge, humans are unique in their use of complex symbolic thinking and creative imagination."

Imagination and creativity are close friends. I am imagining what I will write today. Then I start writing. The result will probably be not quite as I have imagined it.

"You may say I'm a dreamer

But I'm not the only one

I hope someday you'll join us

And the world will be as one"

You may say I'm a dreamer....

Yes, you may say I'm a dreamer. But I'm not the only one. Last night I dreamed about Hilbert Algebra. The dream had its roots in my past memory. My PhD thesis was partly about equilibrium states in quantum thermodynamics, the algebraic approach, with KMS (Kubo-Martin_Schwinger)  states. Tomita and Takesaki, the two Japanese mathematicians, developed the theory of Hilbert algebras and modular automorphisms. Much later, in 1994,  Alain Connes and Carlo Rovelli  applied it  to the "flow of time":

"We consider the cluster of problems raised by the relation between the notion of time, gravitational theory, quantum theory and thermodynamics; in particular, we address the problem of relating the “timelessness” of the hypothetical fundamental general covariant quantum field theory with the “evidence” of the flow of time. "

But now is now, and we are dealing with the Ur atom of space - the geometric Clifford algebra of 3D space Cl(V). We have an algebra. Buit where is "Hilbert"? Searching for "Hilbert Algebra" leads us to Wolfram MathWorld. There we find conditions 1, 2, 3, 4, 5 that need to be satisfied.

In our case only condition 3 counts. All other are automatically satisfied. So, we should have an algebra with involution (we have one), which is a Hilbert space (and Cl(V) is a Hilbert space, as we will see soon),  with a scalar product <a,b> satisfying

<ba,c> = <b,ca*>.                    (1)

We need to define Hilbert space scalar product on Cl(V). We have already done it in Part 11,

where we have defined:

"... But then we have also the second natural "bilinear" form:

B_τ(u,v) =t_n(τ(u)v),

In fact, this form is, when considered as a form on a complex vectors space,  Hermitian: it is complex linear in v, but complex anti-linear in u."

In Part 12 we further wrote:

B_τ(u,u) = |p₀|² + |p|².

This is our Hilbert space scalar product - the scalar part of τ(u)v: we set

<u,v> = the scalar part of the product τ(u)v. If u = (p₀,p), v = (q₀,q), then

<u,v> = p_0^*q₀ + p^*· q.

Then we need an involution in our algebra, we need a *-algebra. We take τ for our involution. We just set u* = τ(u). There may arise some confusion now. Because in my blog I often used the star to denote the complex conjugation. And now I am using the same symbol to denote the conjugation in the algebra! Well, the meaning will be defined by the context, and when confusion can arise, it will have to be explicitly addressed.

Let us recall the action of τ:

If u = (p₀,p), then τ(u) = (p₀*,p*), thus u* =  (p₀*,p*). In other words:

(p₀,p)* = (p₀*,p*).

Not too bad! I suppose we can live with that.

Now comes the crutial point:

Exercise 1. Prove (1).

Hint: there are at least two ways of proving it. Hard way, using definitions, and not-so-hard, if you are creative.